Let A be a 6 X 9 matrix. If Nullity (ATA^TAT) = 2 then Nullity(A) = 2

Question
Matrix transformations
asked 2021-02-11
Let A be a 6 X 9 matrix. If Nullity \(\displaystyle{\left({A}{T}{A}^{{T}}{A}{T}\right)}\) = 2 then Nullity(A) = 2

Answers (1)

2021-02-12
It is given that AA is a 6×9 matrix and Nullity \(\displaystyle{\left({A}^{{T}}\right)}={2}\). Let rank(A)=k. Then we know that rank \(\displaystyle{\left({A}^{{T}}\right)}={k}\). By Rank-nullity theorem we have
rank(A)+Nullity(A)=9andrank(AT)+Nullity(AT)=6
It follows that
rank \(\displaystyle{\left({A}^{{T}}\right)}={6}−{2}={4}  ⟹  {k}={4}\)
Therefore we get
Nullity(A)=9−k=9−4=5
Hence the given statement is False.
0

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